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Conditional mathematical expectation

A function of an elementary event that characterizes the random variable with respect to a certain -algebra. Let be a probability space, let be a real-valued random variable with finite expectation defined on this space and let be a -algebra, . The conditional expectation of with respect to is understood to be a random variable , measurable with respect to and such that

(*)

for each . If the expectation of is infinite (but defined), i.e. only one of the numbers and is finite, then the definition of the conditional expectation by means of (*) still makes sense but may assume infinite values.

The conditional expectation is uniquely defined up to equivalence. In contrast to the mathematical expectation, which is a number, the conditional expectation represents a function (a random variable).

The properties of the conditional expectation are similar to those of the expectation:

1) if, almost certainly, ;

2) for every real ;

3) for arbitrary real and ;

4) ;

5) for every convex function . Furthermore, the following properties specific to the conditional expectation hold:

6) If is the trivial -algebra, then ;

7) ;

8) ;

9) if is independent of , then ;

10) if is measurable with respect to , then .

There is a theorem on convergence under the integral sign of conditional mathematical expectation: If is a sequence of random variables, , and almost certainly, then, almost certainly, .

The conditional expectation of a random variable with respect to a random variable is defined as the conditional expectation of relative to the -algebra generated by .